Deforming a mesh in BGE runtime
The Blender Game Engine and the Blender's simulation system use two different pysics solvers. To see the game physics settings you have to set the Blender engine to 'Blender Game':

But the softbody settings of the game engine do not offer any settings for plastic deformation:


So why is there a difference? 
The solver used in the animation system is more powerfull and accurate, but as you may have noticed it is much too slow for a game.
The solver in the Game Engine on the other hand uses a very fast position-based softbody simulation called Shape Matching (see the paper for more information). While the paper also describes the implementation of plastic deformations, this is either not implemented or not activated in Blender.